JEE Mathematics Properties of Triangles Notes Set 02

Sine, Cosine and Projection Rules

Standard notation. In triangle ABC the sides opposite A, B, C are a, b, c, the semi-perimeter is \(s=\frac{a+b+c}2\), the area is \(\Delta\), the circumradius R and the inradius r.

Sine rule. \(\frac a{\sin A}=\frac b{\sin B}=\frac c{\sin C}=2R\). Cosine rule. \(\cos A=\frac{b^2+c^2-a^2}{2bc}\). Projection rule. \(a=b\cos C+c\cos B\). Napier's analogy. \(\tan\frac{A-B}2=\frac{a-b}{a+b}\cot\frac C2\).

Half-Angle Formulae and Area

\(\sin\frac A2=\sqrt{\frac{(s-b)(s-c)}{bc}}\), \(\cos\frac A2=\sqrt{\frac{s(s-a)}{bc}}\), \(\tan\frac A2=\frac{\Delta}{s(s-a)}\).

Area: \(\Delta=\frac12ab\sin C=\sqrt{s(s-a)(s-b)(s-c)}\) (Heron). The length of the median from A is \(\frac12\sqrt{2b^2+2c^2-a^2}\), and the bisector of angle C has length \(\frac{2ab\cos(C/2)}{a+b}\).

Radii of Circles Connected with a Triangle

\(R=\frac{abc}{4\Delta}=\frac a{2\sin A}\), \(r=\frac\Delta s=4R\sin\frac A2\sin\frac B2\sin\frac C2\), exradii \(r_1=\frac\Delta{s-a}\), \(r_2=\frac\Delta{s-b}\), \(r_3=\frac\Delta{s-c}\).

Useful results: \(\cos A+\cos B+\cos C=1+\frac rR\), \(r_1+r_2+r_3-r=4R\), \(r_1r_2+r_2r_3+r_3r_1=s^2\), \(\frac1r=\frac1{r_1}+\frac1{r_2}+\frac1{r_3}\), \(\Delta^2=rr_1r_2r_3\).

Worked Examples

Example 1. If \(a:b:c=3:7:8\), find \(R:r\). Take sides 3, 7, 8: \(s=9\), \(\Delta=6\sqrt3\), so \(R=\frac{7}{\sqrt3}\), \(r=\frac2{\sqrt3}\) and \(R:r=7:2\).

Example 2. In a triangle with \(\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}\), find \(\cos A:\cos B:\cos C\). Put each ratio equal to k to get \(a=7k\), \(b=6k\), \(c=5k\). The cosine rule gives \(\cos A=\frac15,\ \cos B=\frac{19}{35},\ \cos C=\frac57\), so the ratio is \(7:19:25\).

Example 3. If \(a^2+b^2+c^2=8R^2\) then \(\sin^2A+\sin^2B+\sin^2C=2\), which forces \(\cos A\cos B\cos C=0\): the triangle is right angled.

Quick Tips

Convert every side to a multiple of \(\sin\) of the opposite angle when the question is mixed, and test any proposed identity with a 3-4-5 triangle. For the ambiguous case check that \(\sin B\le1\) before counting triangles.